Problem set 4
... Due at the beginning of class on Tuesday, August 18. Matrix of discretized derivative In the lecture it was mentioned that Newton’s equation ẍ = f could be written as a matrix equation when discretized. Here you will do this for the simpler problem of the first derivative. Given the position of a p ...
... Due at the beginning of class on Tuesday, August 18. Matrix of discretized derivative In the lecture it was mentioned that Newton’s equation ẍ = f could be written as a matrix equation when discretized. Here you will do this for the simpler problem of the first derivative. Given the position of a p ...
Worksheet 9 - Midterm 1 Review Math 54, GSI
... R 1the map I : C(R) → R where C(R) is the space of continuous functions on R given by I(f ) = 0 f (t) dt. Check that I is linear. Is I injective? Is I surjective? Find the dimensions of the kernel and range of I when it is restricted to the supspace of polynomials of degree ≤ n, Pn , of C(R). 16. Fo ...
... R 1the map I : C(R) → R where C(R) is the space of continuous functions on R given by I(f ) = 0 f (t) dt. Check that I is linear. Is I injective? Is I surjective? Find the dimensions of the kernel and range of I when it is restricted to the supspace of polynomials of degree ≤ n, Pn , of C(R). 16. Fo ...
12 How to Compute the SVD
... We saw earlier that the nonzero singular values of A are given by the square roots of the nonzero eigenvalues of either A∗ A or AA∗ . However, computing the singular values in this way is usually not stable (cf. solution of the normal equations). Recall the strategy for finding the eigenvalues of a ...
... We saw earlier that the nonzero singular values of A are given by the square roots of the nonzero eigenvalues of either A∗ A or AA∗ . However, computing the singular values in this way is usually not stable (cf. solution of the normal equations). Recall the strategy for finding the eigenvalues of a ...
Freivalds` algorithm
... Another Randomized Algorithm Freivalds’ Algorithm for Matrix Multiplication ...
... Another Randomized Algorithm Freivalds’ Algorithm for Matrix Multiplication ...
Let X ∈ R n×p denote a data matrix with n observations and p
... Let X ∈ Rn×p denote a data matrix with n observations and p variables with xi = (xi1 , . . . , xip )> for i = 1, . . . , n. We would like to perform fuzzy clustering to attain K clusters. Let uik denote the membership of observation i to cluster k and U the membership matrix, as defined in the lectu ...
... Let X ∈ Rn×p denote a data matrix with n observations and p variables with xi = (xi1 , . . . , xip )> for i = 1, . . . , n. We would like to perform fuzzy clustering to attain K clusters. Let uik denote the membership of observation i to cluster k and U the membership matrix, as defined in the lectu ...
Set 3
... 7. Find a real 2 × 2 matrix A (with A2 6= I and A3 6= I ) so that A6 = I . For your example, is A4 invertible? 8. Let A, B , and C be n × n matrices with A and C invertible. Solve the equation ABC = I − A for B . 9. If a square matrix M has the property that M 4 − M 2 + 2M − I = 0, show that M is i ...
... 7. Find a real 2 × 2 matrix A (with A2 6= I and A3 6= I ) so that A6 = I . For your example, is A4 invertible? 8. Let A, B , and C be n × n matrices with A and C invertible. Solve the equation ABC = I − A for B . 9. If a square matrix M has the property that M 4 − M 2 + 2M − I = 0, show that M is i ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.